Elsevier

Optics Communications

Volume 199, Issues 1–4, 15 November 2001, Pages 65-75
Optics Communications

Phase retrieval from series of images obtained by defocus variation

https://doi.org/10.1016/S0030-4018(01)01556-5Get rights and content

Abstract

We develop and compare three different methods of phase retrieval from series of image measurements obtained at different defocus values. The first approach is an approximate solution to the transport of intensity equation (TIE) based on Fourier transforms. The second is an exact solution of the TIE, using multigrid methods. Lastly an iterative approach, using the free space propagator between image planes, is discussed. The iterative scheme is robust in the presence of discontinuities in the phase, unlike those methods based on solution of the TIE. The performance of the different methods in the presence of noise is discussed. Application of these methods to a set of experimental images taken using X-ray imaging is investigated. A computer program which can reproduce the simulations and analyse experimental image data is briefly discussed.

Introduction

Non-interferometric determination of the phase of quantum mechanical and classical wave fields (i.e. phase retrieval) is a topic of current interest in a number of areas where either phase imaging or structure retrieval is an issue. For example, phase measurement is topical for optical [1], X-ray [2], [3], neutron [4], electron [5], [6], [7], [8] and atom [9] wave fields. In this paper we develop and compare three different methods for phase retrieval from series of images taken at different defocus values. Firstly we consider two approaches to the solution of the transport of intensity equation (TIE), which is based on conservation of flux and has been considered, directly or indirectly, by several authors [10], [11], [12], [13], [14]. Here, in the first instance, we pursue an approximate solution to the TIE, starting out by making an approximation to obtain an equation of the Poisson type for an auxiliary function, as suggested by Teague [10]. We then show that, by differentiation, we can obtain a second Poisson equation for the phase itself, rather than pursuing Teague's approach based on Green functions. All the steps in this approach can be cast in terms of Fourier transforms, using well-known standard results. We note that another algorithm which starts from Teague's approximation is that proposed by Paganin and Nugent [13], which requires use of the calculus of pseudo-differential operators. As an alternative to making any approximations, the TIE can be solved exactly using multigrid methods [15]. We implement such an approach and are thus able to compare the effect of Teague's approximation under various circumstances. Lastly we consider an iterative approach, in the spirit of Gerchberg and Saxton [16] and Misell [17], based on the use of the free space propagator between image planes [18], [19]. A major advantage of this iterative scheme is that it is robust in the presence of discontinuities in the phase, unlike the methods based on solution of the TIE. The performance of the different methods in the presence of noise is discussed. Application of these methods to a set of experimental images taken using X-ray imaging is investigated. A computer program which can reproduce the simulations in this paper and can be used to analyse experimental data is briefly discussed.

Section snippets

Phase retrieval methods

The starting point for our discussion is the Schrödinger equation for the propagation of a wave in free space and in three dimensions,(∇2+k2)Ψ(r)=0,where k is the wave number and is related to the wavelength λ of the radiation by k=2π/λ. Let us assume that the wave function Ψ(r) can be considered as a perturbation of a plane wave travelling along the z direction and can be written in the formΨ(r,z)=exp(ikz)ξ(r,z),where r is a vector in the x–y plane and perpendicular to the z direction. Then

Model solutions of the phase problem for pseudo-data

To investigate the performance of the phase retrieval methods discussed in the previous section we have started from the following model image and phase maps (assumed to be at zero defocus). The image is given byI(x,y)=1.0−0.9exp−b2(x−x1)2+(y−y1)2+exp−b2(x−x2)2+(y−y2)2,where b=0.0027. The location of the centre of the first Gaussian peak is at (x1,y1) with x1=2nx/5 and y1=3ny/5 and nx and ny are the numbers of pixels along the x and y directions respectively. The second Gaussian is centred at (x

Summary and conclusions

We have developed and compared the use of three different methods of phase retrieval from series of image measurements obtained at different defocus values. The first approach was an approximate solution to the TIE based on Fourier transforms. The second method was an exact solution of the TIE using multigrid methods. Finally an iterative approach using the free space propagator between image planes was discussed. The approximate solution of the TIE was adequate in circumstances where the phase

Acknowledgements

The authors would like to thank, Dr. K. Amos, H.M.L. Faulkner, Dr. P.J. McMahon, Prof. K.A. Nugent, Dr. D. Paganin and Prof. H. von Geramb for stimulating and useful discussions. We are particularly grateful to Dr. P.J. McMahon for providing the experimental X-ray images. L.J.A. acknowledges financial support from the Australian Research Council.

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